Boltzmann Equations

The explicit form of the Boltzmann equations for the temperature and polarization follows directly from the Clebsch-Gordan relation of Eqn. (58). For the temperature (s=0),  

$$\dot \Theta_\ell^{(m)} = k\Bigg[ { \, {}_{0}^{\vphantom{m}} ... ...\dot\tau \Theta_\ell^{(m)} + S_\ell^{(m)}, \qquad (\ell \ge m).$$ (52)

The term in the square brackets is the free streaming effect that couples the $\ell$-modes and tells us that in the absence of scattering power is transferred down the hierarchy when $k\eta \agt1$. This transferral merely represents geometrical projection of fluctuations on the scale corresponding to k at distance $\eta$ which subtends an angle given by $\ell \sim k\eta$. The main effect of scattering comes through the $\dot\tau \Theta_\ell^{(m)}$ term and implies an exponential suppression of anisotropies with optical depth in the absence of sources. The source $S_\ell^{(m)}$ accounts for the gravitational and residual scattering effects,  

$$\begin{array} {lll} S_0^{(0)} = \dot\tau \Theta_0^{(0)}- \do... ... \dot H \vphantom{\displaystyle{\dot a \over a}}\, .\end{array}$$ (53)

The presence of $\Theta_0^{(0)}$ represents the fact that an isotropic temperature fluctuation is not destroyed by scattering. The Doppler effect enters the dipole $(\ell=1)$equation through the baryon velocity v_B^(m) term. Finally the anisotropic nature of Compton scattering is expressed through  

$$P^{(m)} = {1 \over 10} \left[ \Theta_2^{(m)} - \sqrt{6} E_2^{(m)} \right],$$ (54)

and involves the quadrupole moments of the temperature and E-polarization distribution only.

The polarization evolution follows a similar pattern for $\ell \ge 2$, $m\ge 0$ from Eqn. (58) with $s=\pm 2$,  

$$\begin{array} {rcl}\displaystyle{}\dot E_\ell^{(m)} &=& k ... ... B_{\ell + 1}^{(m)} \Bigg] - \dot\tau B_\ell^{(m)} .\end{array}$$ (55)

Notice that the source of polarization P^(m) enters only in the E-mode quadrupole due to the opposite parity of $\Theta_2$and B₂. However, as discussed in §IIB, free streaming or projection couples the two parities except for the m=0 scalars. Thus $B_\ell^{(0)}=0$ by geometry regardless of the source. It is unnecessary to solve separately for the m=-|m| relations since they satisfy the same equations and solutions with $B_\ell^{(-\vert m\vert)} = -B_\ell^{(\vert m\vert)}$ and all other quantities equal.

To complete these equations, we need to express the evolution of the metric sources $(\Phi,\Psi,V,H)$. It is to this subject we now turn.